In this paper, potential‐dependent transformations are used to transform the four‐component Dirac Hamiltonian to effective two‐component regular Hamiltonians. To zeroth order, the expansions give second order differential equations (just like the Schrodinger equation), which already contain the most important relativistic effects, including spin–orbit coupling. One of the zero order Hamiltonians is identical to the one obtained earlier by Chang, Pelissier, and Durand [Phys. Scr. 34, 394 (1986)]. Self‐consistent all‐electron and frozen‐core calculations are performed as well as first order perturbation calculations for the case of the uranium atom using these Hamiltonians. They give very accurate results, especially for the one‐electron energies and densities of the valence orbitals.

Relativistic regular two‐component Hamiltonians

After a discussion of the problems associated with the non-relativistic limit of the Dirac equation and of the expansion of the exact eigenvalues and eigenfunctions of the H atom in powers ofc−2 the traditional approaches for a perturbation theory of relativistic effects are critically reviewed. Then a direct perturbation theory is presented, that is characterized by a change of the metric in 4-component spinor space such that the Levy-Leblond equation appears as the straightforward non-relativistic limit of the Dirac equation. The various orders in perturbation theory of the energy and the wave function are derived first in a direct way, then in a resolvent formalism. The formulas are very compact and easily generalizeable to arbitrary order. All integrals that arise to any order exist, and no controlled cancellation of divergent terms (as in other approaches) is necessary. In the same philosophy an iterative approach towards the solution of the Dirac equation is derived, in which the solution of the Schrodinger equation is the first iteration step.

Perturbation theory of relativistic corrections

Effective Hamiltonians and Pseudo‐Operators as Tools for Rigorous Modelling

The general problem of relativistic corrections to the kinetic energy in quasirelativistic theories, is discussed and related formulas are developed. It is shown that the well-known mass-velocity operator, Hmv = (−α2/8)p4, is incorrect and does not provide any proper relativistic corrections.

Invalidity of the ubiquitous mass‐velocity operator in quasirelativistic theories

The accurate ab initio determination of inter-ionic potentials requires the consideration of four important factors. These are relativistic effects for ions of high atomic number, the environmentally induced modifications of ion wavefunctions, the damping of reliably calculated inter-ionic dispersive attractions and the avoidance of density functional descriptions of the uncorrelated potentials. The relativistic modifications of the behaviour of valence electrons in elements heavier than those of the third series of transition elements are too large to be treated by first order perturbation theory. The relativistic integrals programme (RIP), which works directly with the Dirac equation and four component wavefunctions for the individual electrons, is used to compute fully relativistic inter-ionic potentials. The mechanisms through which ion wavefunctions are modified by their environment in the crystal are discussed. A simple yet physically realistic model for computing such wavefunctions is pres...

Four Important Factors in the ab initio Determination of Accurate Inter-Ionic Potentials

The use of closed form operators and the corresponding power series expanded operators in Foldy-Wouthuysen transformations is discussed. It is shown that in general different transformed hamiltonians arise. This is demonstrated using the free electron part of the Dirac hamiltonian; the operators and their expectation values using hydrogenic functions are considered.

On the expectation values of relativistic corrections to the Hamiltonian

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On the expectation values of relativistic corrections to the Hamiltonian